A nonlinear scenario for development of vortex layer instability in gravity field

A Hamiltonian version of contour dynamics is formulated for models of constant-vorticity plane flows with interfaces. The proposed approach is used as a framework for a nonlinear scenario for instability development. Localized vortex blobs are analyzed as structural elements of a strongly perturbed wall layer of a vorticity-carrying fluid with free boundary in gravity field. Gravity and vorticity effects on the geometry and velocity of vortex structures are examined. It is shown that compactly supported nonlinear solutions (compactons) are candidates for the role of particle-like vortex structures in models of flow breakdown. An analysis of the instability mechanism demonstrates the possibility of a self-similar collapse. It is found that the vortex shape stabilizes at the final stage of the collapse, while the vortex sheet strength on its boundary increases as (t ₀ ? t)^?1, where t ₀ is the collapse time.     

Some exact solutions of the local induction equation for the motion of a vortex in a Bose–Einstein condensate with a Gaussian density profile

The dynamics of a vortex filament in a Bose–Einstein condensate whose equilibrium density in the reference frame rotating at the angular velocity Ω is Gaussian with the quadratic form r·D?r has been considered. It has been shown that the equation of motion of the filament in the local-induction approximation permits a class of exact solutions in the form R(β, t) = βM(t) + N(t) of a straight vortex, where β is the longitudinal parameter and is the time. The vortex slips over the surface of an ellipsoid, which follows from the conservation laws N · D?N=C ₁ and M · D?N=C ₀=0. The equation of the evolution of the tangential vector M(t) appears to be closed and has integrals of motion M ·D?M=C ₂ and (|M| ? M· G?Ω) = C, with the matrix G? = 2(I?TrD? ? D?)^?1. Crossing of the respective isosurfaces specifies trajectories in the phase space.     

Self-similar magnetic structures and magnetic flux “Giant” creep

The penetration of a magnetic flux into a type-II high-T _c superconductor occupying the half-space x > 0 is considered. At the superconductor surface, the magnetic field amplitude increases in accordance with the law b(0, t) = b ₀(1 + t)^m (in dimensionless coordinates), where m > 0. The velocity of penetration of vortices is determined in the regime of thermally activated magnetic flux flow: v = v ₀exp?ub;?(U _0/T )(1-b?b/?x)?ub;, where U ₀ is the effective pinning energy and T is the thermal energy of excited vortex filaments (or their bundles). magnetic flux “Giant” creep (for which U ₀/T? 1) is considered. The model Navier-Stokes equation is derived with nonlinear “viscosity” v ∝ U ₀/T and convection velocity v _f ∝ (1 ? U ₀/T). It is shown that motion of vortices is of the diffusion type for j → 0 (j is the current density). For finite current densities 0 < j < j _c, magnetic flux convection takes place, leading to an increase in the amplitude and depth of penetration of the magnetic field into the superconductor. It is shown that the solution to the model equation is finite at each instant (i.e., the magnetic flux penetrates to a finite depth). The penetration depth x _eff ^A (t) ∝ (1 + t)^{(1 + m/2)/2} of the magnetic field in the superconductor and the velocity of the wavefront, which increases linearly in exponent m, exponentially in temperature T, and decreases upon an increase in the effective pinning barrier, are determined. A distinguishing feature of the solutions is their self-similarity; i.e., dissipative magnetic structures emerging in the case of giant creep are invariant to transformations b(x, t) = β^m b(t/β, x/β^{(1 + m/2)/2}), where β > 0.     

Impurity transport in fractal media in the presence of a degrading diffusion barrier

We have analyzed the transport regimes and the asymptotic forms of the impurity concentration in a randomly inhomogeneous fractal medium in the case when an impurity source is surrounded by a weakly permeable degrading barrier. The systematization of transport regimes depends on the relation between the time t ₀ of emergence of impurity from the barrier and time t _* corresponding to the beginning of degradation. For t ₀ < t _*, degradation processes are immaterial. In the opposite situation, when t ₀ > t _*, the results on time intervals t < t _* can be formally reduced to the problem with a stationary barrier. The characteristics of regimes with t _* < t < t ₀ depend on the scenario of barrier degradation. For an exponentially fast scenario, the interval t _* < t < t ₀ is very narrow, and the transport regime occurring over time intervals t < t _* passes almost jumpwise to the regime of the problem without a barrier. In the slow power-law scenario, the transport over long time interval t _* < t < t ₀ occurs in a new regime, which is faster as compared to the problem with a stationary barrier, but slower than in the problem without a barrier. The asymptotic form of the concentration at large distances from the source over time intervals t < t ₀ has two steps, while for t > t ₀, it has only one step. The more remote step for t < t ₀ and the single step for t > t ₀ coincide with the asymptotic form in the problem without a barrier.     

The reactions N14(n, α)B11 and N14(n,t)C12 observed with a gridded ionization chamber

The cross sections for the reactions N¹⁴(n, α)B¹¹ and N¹⁴(n, t)C¹² have been measured in the neutron energy range 4.0 to 6.4 MeV and at 2.5 MeV. Mono-energetic neutrons were produced in the D(d, n) He³ reaction using a gas target. The (n, α) and (n, t) disintegrations were detected in a gridded ionization chamber filled with an argonnitrogen mixture. The response of the chamber under different operation conditions is described. The excitation functions, measured with a neutron energy resolution of 40 to 50 keV, are given for theα ₀ group from the N¹⁴(n,α)B¹¹ reaction over the entire neutron energy range and for theα ₁ group and the t₀ group from N¹⁴(n, t) C¹² for neutron energies above 4.3 and 5.6 MeV, respectively.     

Gamow Vectors and Borel Summability in a Class of Quantum Systems

We analyze the detailed time dependence of the wave function ψ(x,t) for one dimensional Hamiltonians \(H=-\partial_{x}^{2}+V(x)\) where V (for example modeling barriers or wells) and ψ(x,0) are compactly supported.We show that the dispersive part of ψ(x,t) is the Borel sum of its asymptotic series in powers of t ^?1/2, t→∞. The remainder, the difference between ψ and the Borel sum, i.e., the exponential part of the transseries of ψ, is a convergent expansion of the form \(\sum_{k=0}^{\infty}g_{k}\Gamma_{k}(x)e^{-\gamma_{k} t}\), where Γ _k are the Gamow vectors of H, and iγ _k are the associated resonances; generically, all g _k are nonzero. For large k, γ _k ～const?klog?k+k ² π ² i/4. The effect of the Gamow vectors is visible when time is not very large, and the decomposition defines rigorously resonances and Gamow vectors in a nonperturbative regime, in a physically relevant way.The decomposition allows for calculating ψ for moderate and large t, to any prescribed exponential accuracy, using optimal truncation of power series plus finitely many Gamow vectors contributions.The analytic structure of ψ is perhaps surprising: in general (even in simple examples such as square wells), ψ(x,t) turns out to be C ^∞ in t but nowhere analytic on ?⁺. In fact, ψ is t-analytic in a sector in the lower half plane and has the whole of ?⁺ a natural boundary. In the dual space, we analyze the resurgent structure of ψ.     

Long-time relaxation phenomena in Pb<Subscript>0.94</Subscript>Ba<Subscript>0.06</Subscript>Sc<Subscript>0.5</Subscript>Nb<Subscript>0.5</Subscript>O<Subscript>3</Subscript> (PBSN-6) single crystals

Long-time polarization relaxation in the temperature range where PBSN-6 single crystals reside in the relaxor state was studied. An analysis of the time dependence of the permittivity ε′(t) performed at measuring frequencies from 1 Hz to 1 kHz in weak electric fields E₀ showed that the relaxation (or freezing) times derived by extrapolating relations of the type ε′(t) ～ log(t/t₀) and ε′(t) ～ exp{?[ln(t/t₀)]_β} range from 10⁸ to 10¹¹ min and depend substantially on the bias voltage applied to the sample. A study of the pattern of the dielectric response in moderate and strong infralow-frequency fields revealed that, after a sample was maintained under a bias lower than the coercive force, it no longer exhibited the additional anomalies in the amplitude dependences of the effective loss tangent tan_eff(E₀) than were observed in a thermally recuperated sample.     

Ferromagnetism of metallic compounds with the perovskite structure

On the basis of the concept of strong interaction in one unit cell, it is stated that ferromagnetic instability can occur in a system with hops between oxygen anions and transition metal (Me) cations in A_1?xB_xMeO₃ compounds. A phase diagram is constructed to describe the occurrence of ferromagnetic ordering as a function of the average number of holes (h_{t, d}) and (h_p) in the low-spin or high-spin t _2g ⁶ or high-spin 3d¹⁰ shell of the transition metal and in the 2p⁶ shells of O^2?.     

Long-range spin correlations in a honeycomb spin model with a magnetic field

We consider the spin-1/2 model on the honeycomb lattice [A. Kitaev, Ann. Phys. 321, 2 (2006)] in the presence of a weak magnetic field h _α ? J. Such a perturbation treated in the lowest nonvanishing order over h _α leads [K.S. Tikhonov, M.V. Feigel’man, and A.Yu. Kitaev, Phys. Rev. Lett. 106, 067203 (2011)] to a powerlaw decay of irreducible spin correlations 《s ^z (t, r)s ^z (0, 0)》 ∝ h _z ² f(t, r), where f(t, r) ∝ [max(t, Jr)]^–4. We have studied the effects of the next order of perturbation in h _z and found an additional term of the order h _z ⁴ in the correlation function 《s ^z (t, r)s ^z (0, 0)》 which scales as h _z ⁴ cosγ/r ³ at Jt? r, where γ is the polar angle in the 2D plane. We demonstrate that such a contribution can be understood as a result of a perturbation of the effective Majorana Hamiltonian by the weak imaginary vector potential A _x ∝ i h _z ² .     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

We study the asymptotic structure of the first K largest eigenvalues λ _k,V and the corresponding eigenfunctions ψ(?;λ _k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ? ^ν , provided the distribution function F(?) of i.i.d. potential ξ(?) satisfies condition ?log(1?F(t))=o(t ³) and some additional regularity conditions as t→∞. For z∈V, denote by λ ⁰(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ _V +ξ(z)δ _z in l ²(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ ⁰(?) in V. We first show that the eigenvalues λ _k,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ _k,V ?B _V )a _V , where the normalizing constants a _V >0 and B _V are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(?;λ _k,V ) is shown to be asymptotically completely localized (as V↑?) at the sites z _k,V ∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.     

Collapse of vortex lines in hydrodynamics

A new mechanism is proposed for collapse in hydrodynamics associated with the “breaking” of vortex lines. The collapse results in the formation of point singularities of the vorticity field, i.e., a generalized momentum curl. At the point of collapse the vorticity |Ω| increases as ((t ₀ ? t)^?1 and its spatial distribution for t → t ₀ approaches quasi-two-dimensional: in the “soft” direction contraction obeys the law l ₁ → (t ₀ ? t)^3/2 whereas in the other two “hard” directions it obeys l ₂ → (t ₀ ? t)^1/2. It has been shown that this collapse scenario takes place in the general case for three-dimensional integrable hydrodynamics with the Hamiltonian ? = ∫|Ω| d r.     

Nonself-similar regimes of isothermal collapse of protostellar clouds

We consider the development of inhomogeneity in the isothermal collapse of protostellar clouds. The initial and boundary conditions correspond to the classical statement of the problem on the contraction of a homogeneous cloud from a given volume. A centered rarefaction wave is shown to propagate from the outer boundary of the cloud toward its center at the first collapse stage. Analysis reveals two possible regimes of isothermal collapse, depending on the relationship between the rarefaction wave focusing time t^* and the cloud free-fall collapse time t_ff. For cold clouds, t^*=t _ff and the rarefaction wave is not reflected. In this case, as time elapses, the cloud collapse becomes self-similar with the characteristic density profile ρ～r^?2. In hot clouds, t*<t _ff and the focusing can take place before the formation of an opaque core. Since the velocities of the rarefaction wave along and across magnetic field lines in a magnetized cloud are different, its front assumes a shape elongated along magnetic field lines. Depending on the initial conditions, based on analytical estimates, we investigate various possible scenarios for the collapse of magnetic protostellar clouds.     

<Emphasis Type="Italic">Z</Emphasis><Superscript>0</Superscript>-boson decays in a strong electromagnetic field

The probability of Z ⁰-boson decay to a pair of charged fermions in a strong electromagnetic field, Z ⁰ → \(\bar f\) f, is calculated. On the basis of a method that employs exact solutions to relativistic wave equations for charged particles, an analytic expression for the partial decay width Γ(?) = Γ(Z ⁰ → \(\bar f\) f) is obtained at an arbitrary value of the parameter ? = \(eM_Z^{ - 3} \sqrt { - (F_{\mu \nu } q^\nu )^2 } \), which characterizes the external-field strength. The total Z ⁰-boson decay width in an intense electromagnetic field, Γ _Z (?), is calculated by summing these results over all known generations of charged leptons and quarks. It is found that, in the region of relatively weak fields (? < 0.06), the field-induced corrections to the standard Z ⁰-boson decay width in a vacuum do not exceed 2%. As ? increases, the total decay width Γ _Z (?) develops oscillations against the background of its gradual decrease to the absolute-minimum point. At ?_min = 0.445, the total Z ⁰-boson decay width reaches the minimum value of Γ _Z (?_min) = 2.164 GeV, which is smaller than the Z ⁰-boson decay width in a vacuum by more than 10%. In the region of superstrong fields (? > 1), Γ _Z (?) grows monotonically with increasing external-field strength. In the region ? > 5, the t-quark-production process Z ⁰ → \(\bar t\) t, which is forbidden in the absence of an external field, begins contributing significantly to the total decay width of the Z ⁰ boson.     

Resolvent Expansion and Time Decay of the Wave Functions for Two-Dimensional Magnetic Schrödinger Operators

We consider two-dimensional Schrödinger operators H(B, V) given by Eq. (1.1) below. We prove that, under certain regularity and decay assumptions on B and V, the character of the expansion for the resolvent (H(B, V) ? λ)^?1 as λ → 0 is determined by the flux of the magnetic field B through \({\mathbb{R}^2}\) . Subsequently, we derive the leading term of the asymptotic expansion of the unitary group e ^{?i t H(B, V)} as t → ∞ and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ^{?i t H(0, V)}.     

Stable and Unstable Vortex Knots in a Trapped Bose Condensate

The dynamics of a quantum vortex toric knot T_P,Q and other analogous knots in an atomic Bose condensate at zero temperature in the Thomas–Fermi regime is considered in the hydrodynamic approximation. The condensate has a spatially inhomogeneous equilibrium density profile ρ(z, r) due to the action of an external axisymmetric potential. It is assumed that z_*= 0, r_*= 1 is the point of maximum of function rρ(z, r), so that δ(rρ) ≈ –(α–)z²/2–(α + )(δr)²/2 for small z and δr. The geometrical configuration of a knot in the cylindrical coordinates is determined by a complex 2πP-periodic function A(?, t) = Z(?, t) + i[R(?, t))–1]. When |A| ? 1, the system can be described by relatively simple approximate equations for P rescaled functions \({W_n}(\varphi ) \propto A(2\pi n + \varphi ):i{W_{n,t}} = - ({W_{n,\varphi \varphi }} + \alpha {W_n} - \in W_n^*)/2 - \sum\nolimits_{j \ne n} {1/(W_n^* - W_j^*)} \). For = 0, examples of stable solutions of type W _n = θ _n (?–γt)exp(–iωt) with a nontrivial topology are found numerically for P = 3. In addition, the dynamics of various unsteady knots with P = 3 is modeled, and the tendency to the formation of a singularity over a finite time interval is observed in some cases. For P = 2 and small ≠ 0, configurations of type W₀–W₁ ≈ B₀exp(iζ) + C(B₀, α)exp(–iζ) + D(B₀, α)exp(3iζ), where B₀ > 0 is an arbitrary constant, ζ = k₀?–Ω₀t + ζ0, k₀ = Q/2, and Ω₀ = (–α)/2–2/B₀², which rotate about the z axis, are investigated. Wide stability regions for such solutions are detected in the space of parameters (α, B₀). In unstable zones, a vortex knot may return to a weakly excited state.     

Resistive sensor of weak magnetic fields on the basis of a thick HTSC film

The following parameters have been obtained for a thick (thickness t ～50 μm) film of high-temperature superconducting ceramics of the Bi-2223 system have been implemented: magnetosensitivity S _u ～ 29 V T^?1 and resolutions δB ～ 3 nT and δ? ～ 4?₀ in magnetic field and magnetic flux, respectively. It is shown that the film magnetosensitivity can be significantly increased due to the size effect. The expected characteristics, estimated with allowance for the size effect, are S _u ≥ 1000 V T^?1, δB ～ 0.01 nT, δ? ～ 0.01 ?₀, and the range of dynamic measurement ≥150 dB.     

Investigation of zero-frequency solutions to the pion dispersion equation

The instability of nuclear matter is considered for the case where it is generated by the vanishing of the frequencies of collective excitations belonging to specific types (specifically, excitations that have the pion quantum numbers J ^π = 0^?). The behavior of zero-frequency solutions to the pion dispersion equation is analyzed versus the strength G′ of spin—isospin particle—hole interaction. It is shown that there exists a strength value G′_tr (|G′_tr| ? 1) such that, for G′ < G′_tr, zero-frequency solutions are excitations of the ω _P type, while, for G′ ≥ G′_tr, such solutions are excitations of the ω _c type. Excitations of the ω _P type for G′ < ?1 describe the instability of nuclear matter against small density fluctuations (Pomeranchuk’s instability), while excitations of the ω _c type are responsible for the instability associated with pion condensation at G′ ≈ 2. For stable nuclear matter, the solutions ω _P(κ) and ω _c (κ) lie on unphysical sheets of the complex plane of frequency.     

Thermodynamic study of compounds of the Nd-Ba-Cu-O system

Standard enthalpies of formation for solid solutions of composition Nd_{1 + x} Ba_{2 ? x} Cu₃O _y (x = 0–0.8, y = 6.65–7.24) from oxides were determined by solution calorimetry. The heat capacity of NdBa₂Cu₃O_6.87 phase was measured in the range 5–320 K by low-temperature adiabatic calorimetry. The absolute entropy S ^o(T), the difference of enthalpies H ^o(T)-H ^o(0 K), and the reduced Gibbs energy Φ^o(T) = S ^o(T)–[H ^o(T)–H ^o(0)]/T were calculated on the basis of smoothed dependence C _p (T) in the 0–320 K range. An assessment was made for the heat capacities and the absolute entropies of solid solutions Nd_1+x Ba_2?x Cu₃O _y . The obtained set of thermodynamic parameters can be used for the calculation of phase equilibria in the Nd-Ba-Cu-O system.